How to Find the Wind Component Parallel to a Dash Vector?

• -EquinoX-
In summary, to find the component of the wind speed vector w that is parallel to the velocity vector v, we can use the formula w . v/||v||. This will give us the projection of w in the direction of v.
-EquinoX-

Homework Statement

A 100 meter dash is run on a track in the direction of the vector v = 4i + 7j. The wind velocity w is 5i + j km/hr. The rules say that a legal wind speed measured in the direction of the dash must not exceed 5 km/hr.

Find the component of w which is parallel to v.

The Attempt at a Solution

I have no idea to solve this problem

Last edited:
-EquinoX- said:

Homework Statement

A 100 meter dash is run on a track in the direction of the vector v = 4i + 7j. The wind velocity w is 5i + j km/hr. The rules say that a legal wind speed measured in the direction of the dash must not exceed 5 km/hr.

Find the component of w which is parallel to v.

The Attempt at a Solution

I have no idea to solve this problem
You must have some idea. Which operation gives to the component of one vector that is parallel to another?

it's parallel to one another if it's angle between them is 0 right?

so therefore using the geometric of vectors:

v . w = |v| * |w|, right?

-EquinoX- said:
it's parallel to one another if it's angle between them is 0 right?

so therefore using the geometric of vectors:

v . w = |v| * |w|, right?

Sure, if v . w = |v| * |w| then w and v are parallel; but that's not what Hootenanny was asking.

Suppose you want to find 'the component of w' that is parallel to v...how would you do that?

-EquinoX- said:
it's parallel to one another if it's angle between them is 0 right?

so therefore using the geometric of vectors:

v . w = |v| * |w|, right?
That is correct. However, it is perhaps more useful to note that:

$$\mathbf{v}\cdot\left(\frac{\mathbf{w}}{\left\|\mathbf{w}\right\|}\right) = \left|\mathbf{v}\right|\cos\theta$$

That is, if u is a unit vector then the scalar product v.u gives the projection of v in the direction u.

Edit:
gabbagabbahey said:
but that's not what Halls was asking.

hmmm...so how do I apply this to the question?

gabbagabbahey said:
but that's not what Hootenanny was asking.

Is this question asking for a new wind speed/vector which is parallel to v

-EquinoX- said:
Is this question asking for a new wind speed/vector which is parallel to v
No. The question is asking for the component of w that is parallel to v. Can you use the hints I have you in post number 5 to solve this problem?

Hootenanny said:
No. The question is asking for the component of w that is parallel to v. Can you use the hints I have you in post number 5 to solve this problem?

No, I don't...

-EquinoX- said:
No, I don't...
I don't know how I can make it clearer without explicitly giving you the answer. Pay particular attention the the last paragraph in on of my previous posts.
Hootenanny said:
That is correct. However, it is perhaps more useful to note that:

$$\mathbf{v}\cdot\left(\frac{\mathbf{w}}{\left\|\mathbf{w}\right\|}\right) = \left|\mathbf{v}\right|\cos\theta$$

That is, if u is a unit vector then the scalar product v.u gives the projection of v in the direction u.

what is u related to my question?

why is it v . w/||W|| not w . v/||v||

-EquinoX- said:
why is it v . w/||W|| not w . v/||v||

v . w/||W|| gives the component of v parallel to w.

w . v/||v|| gives the component of w parallel to v.

Since you are trying to find the component of the wind speed w parallel to the velocity v; you will indeed want to use w . v/||v|| for your problem.

gabbagabbahey said:
v . w/||W|| gives the component of v parallel to w.

w . v/||v|| gives the component of w parallel to v.

Since you are trying to find the component of the wind speed w parallel to the velocity v; you will indeed want to use w . v/||v|| for your problem.

ok thanks! I got it now

1. What is a vector?

A vector is a mathematical object that has both magnitude (size) and direction. It is often represented graphically as an arrow pointing in a specific direction with a specific length.

2. How is velocity different from speed?

Velocity is a vector quantity that describes both the speed and direction of an object's motion. Speed, on the other hand, is a scalar quantity that only describes how fast an object is moving without considering direction.

3. What is the difference between velocity and acceleration?

Velocity is the rate of change of an object's position with respect to time, while acceleration is the rate of change of an object's velocity with respect to time. In other words, acceleration describes how an object's velocity is changing over time.

4. Can vectors be parallel?

Yes, vectors can be parallel if they have the same direction or are in the same line. Two vectors that are parallel always have the same slope or gradient.

5. How do you calculate the parallel component of a vector?

To calculate the parallel component of a vector, you can use the dot product formula: A · B = |A||B|cosθ, where A and B are two vectors and θ is the angle between them. The parallel component can be found by multiplying the magnitude of one vector with the cosine of the angle between the two vectors.

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